The Spinor Representation of Minimal Surfaces

Abstract

The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield surfaces critical for the Möbius invariant squared mean curvature functional $W$. On the other hand, all $W$-critical spheres and real projective planes arise this way. Thus we determine at the same time the moduli spaces of $W$-critical spheres and real projective planes via the spinor representation.

63 pages, dvi file only, earlier version is GANG preprint III.27 available via http://www.gang.umass.edu/

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